closed graph(数学/泛函分析)指“闭图”:对一个函数或算子 \(T\),把所有形如 \((x, T(x))\) 的点组成的集合称为它的“图像(graph)”;如果这个集合在相应的空间里是闭集,就说 \(T\) 有闭图。该概念常用于闭图定理(Closed Graph Theorem)等结果中。
(在不同语境里 graph 也可指“图(图论)/图表”,但“closed graph”最常见用法是上述“闭图”。)
/ˌkloʊzd ˈɡræf/
A linear operator with a closed graph is often easier to control analytically.
具有闭图的线性算子在分析上往往更容易处理。
In functional analysis, the closed graph theorem implies that a linear operator between Banach spaces is continuous if its graph is closed.
在泛函分析中,闭图定理表明:在巴拿赫空间之间,如果一个线性算子的图像是闭的,那么该算子是连续的。
closed 来自古法语/拉丁语系词根,基本义是“关上的、封闭的”,在数学里引申为“闭的(闭集意义上的)”。graph 源自希腊语 graphein(“书写、描画”),在数学中发展出“图像/点对集合(函数或算子的 graph)”这一含义;合起来 closed graph 即“闭(的)图像/闭图”。